Commit 4278f250 by Zdenek Dvorak

### The treewidth-fragility part.

parent 8c0aa411
 ... ... @@ -7,12 +7,11 @@ \usepackage{epsfig} \usepackage{url} \newcommand{\GG}{{\cal G}} \newcommand{\HH}{{\cal H}} \newcommand{\CC}{{\cal C}} \newcommand{\OO}{{\cal O}} \newcommand{\PP}{{\cal P}} \newcommand{\RR}{{\cal R}} \newcommand{\col}{\text{col}} \newcommand{\vol}{\text{vol}} \newcommand{\eps}{\varepsilon} \newcommand{\mc}[1]{\mathcal{#1}} ... ... @@ -20,6 +19,8 @@ \newcommand{\bb}[1]{\mathbb{#1}} \newcommand{\brm}[1]{\operatorname{#1}} \newcommand{\cbdim}{\brm{dim}_{cb}} \newcommand{\tw}{\brm{tw}} \newcommand{\vol}{\brm{vol}} %%%%% ... ... @@ -449,6 +450,126 @@ certainly can be improved). The dependence of the comparable box dimension on t established is not explicit, as Theorem~\ref{thm-prod} is based on the structure theorem of Robertson and Seymour~\cite{robertson2003graph}. It would be interesting to prove this part of Corollary~\ref{cor-minor} without using the structure theorem. \section{Fractional treewidth-fragility} Suppose $G$ is a connected planar graph and $v$ is a vertex of $G$. For an integer $k\ge 2$, give each vertex at distance $d$ from $v$ the color $d\bmod k$. Then deleting the vertices of any of the $k$ colors results in a graph of treewidth at most $3k$. This fact (which follows from the result of Robertson and Seymour~\cite{rs3} on treewidth of planar graphs of bounded radius) is (in the modern terms) the basis of Baker's technique~\cite{baker1994approximation} for design of approximation algorithms. However, even quite simple graph classes (e.g., strong products of three paths~\cite{gridtw}) do not admit such a coloring (where the removal of any color class results in a graph of bounded treewidth). However, a fractional version of this coloring concept is still very useful in the design of approximation algorithms~\cite{distapx} and applies to much more general graph classes, including all graph classes with strongly sublinear separators and bounded maximum degree~\cite{twd}. We say that a class of graphs $\GG$ is \emph{fractionally treewidth-fragile} if there exists a function $f$ such that for every graph $G\in\GG$ and integer $k\ge 2$, there exist sets $X_1, \ldots, X_m\subseteq V(G)$ such that each vertex belongs to at most $m/k$ of them and $\tw(G-X_i)\le f(k)$ for every $i$ (equivalently, there exists a probability distribution on the set $\{X\subseteq V(G):\tw(G-X)\le f(k)\}$ such that $\text{Pr}[v\in X]\le 1/k$ for each $v\in V(G)$). For example, the class of planar graphs is (fractionally) treewidth-fragile, since we can let $X_i$ consist of the vertices of color $i-1$ in the coloring described at the beginning of the section. Our main result is that all graph classes of bounded comparable box dimension are fractionally treewidth-fragile. We will show the result in a more general setting, motivated by concepts from~\cite{subconvex} and by applications to related representations. The argument is motivated by the idea used in the approximation algorithms for disk graphs by Erlebach et al.~\cite{erlebach2005polynomial}. For a measurable set $A\subseteq \mathbb{R}^d$, let $\vol(A)$ denote the Lebesgue measure of $A$. For two measurable subsets $A$ and $B$ of $\mathbb{R}^d$ and a positive integer $s$, we write $A\sqsubseteq_s B$ if for every $x\in B$, there exists a translation $A'$ of $A$ such that $\vol(A'\cap B)\ge \tfrac{1}{s}\vol(A)$. Note that for two boxes $A$ and $B$, we have $A\sqsubseteq_1 B$ if and only if $A\sqsubseteq B$. An \emph{$s$-comparable envelope representation} $(\iota,\omega)$ of a graph $G$ in $\mathbb{R}^d$ consists of two functions $\iota,\omega:V(G)\to 2^{\mathbb{R}^d}$ such that for some ordering $v_1$, \ldots, $v_n$ of vertices of $G$, \begin{itemize} \item for each $i$, $\omega(v_i)$ is a box, $\iota(v_i)$ is a measurable set, and $\iota(v_i)\subseteq \omega(v_i)$, \item if $i1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $b_{a,j}=1$, implying $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$. \item If $a>1$ and $\ell_{a,j}>\ell_{a-1,j}$, then $\ell_{a-1,j}\ge 2ksd|\omega(v_a)[j]|$ and $$\ell_{a,j}=\frac{\ell_{a-1,j}}{\lfloor \frac{\ell_{a-1,j}}{ksd|\omega(v_a)[j]|}\rfloor}<\tfrac{3}{2}ksd|\omega(v_a)[j]|.$$ \end{itemize} Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$. For any $v_i\in \beta(C)\setminus\{v_a\}$, we have $i\le a$ and $\iota(v_i)\cap C\neq\emptyset$, and since $\omega(v_a)\sqsubseteq_s \iota(v_i)$, there exists a translation $B_i\subseteq C'$ of $\omega(v_a)$ such that $\vol(B_i\cap\iota(v_i))\ge \tfrac{1}{s}\vol(\omega(v_a))$. Since the representation has thickness at most $t$, \begin{align*} \vol(C')&\ge \vol\left(C'\cap \bigcup_{v_i\in \beta(C)\setminus\{v_a\}} \iota(v_i)\right)\\ &\ge \vol\left(\bigcup_{v_i\in \beta(C)\setminus\{v_a\}} B_i\cap\iota(v_i)\right)\\ &\ge \frac{1}{t}\sum_{v_i\in \beta(C)\setminus\{v_a\}} \vol(B_i\cap\iota(v_i))\\ &\ge \frac{\vol(\omega(v_a))(|\beta(C)|-1)}{st}. \end{align*} Since $\vol(C')=(2ksd+2)^d\vol(\omega(v_a))$, it follows that $$|\beta(C)|-1\le (2ksd+2)^dst=f(k),$$ as required. \end{proof} \subsection*{Acknowledgments} This research was carried out at the workshop on Geometric Graphs and Hypergraphs organized by Yelena Yuditsky and Torsten Ueckerdt in September 2021. We would like to thank the organizers and all participants for creating a friendly and productive environment. ... ...
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